A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Decomposition of Poisson process

If $N(t)$ denotes the total number of visitors in the interval $[0,t]$. We suppose that $\lbrace N(t),t > 0 \rbrace $ is a Poisson process with rate $\lambda = 10$ per hour, and that we have $2$ ...
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Problem with infinite product measures

Given some measurable space $\left(X,\mathcal{F}\right)$ and two probability measures $\mu$ and $\nu$ on this space one can define ...
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9 views

Conditional expectation of compound poisson process

The stochastic process $\lbrace Y(t),t > 0 \rbrace$ is a compound Poisson process defined by : $$Y(t)= \sum_{k=1}^{N(t)} X_{k} $$ and $Y(t) = 0$ if ($N(t) = 0)$ where $X_{k} $ has a geometric ...
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What does the “empirical” autocovariance function represent?

My professor gave me the sequence ${X_n} = {1,5,5,1,5,5,...}$ and asked us to compute the empirical autocovariance function given below. $$\displaystyle \hat \rho(1) = \lim_{N \to \infty} \frac{1}{N} ...
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Is the autocovariance function of a sequence identically zero if the sequence is iid?

My professor gives the following definition for the autocovariance function. $$\rho(i,j) = Cov(X_i , X_j)$$$\\$If I have a sequence that is iid, when i compute $\rho(n,n+1)$ for $n \geq 0$, I found ...
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Clark-Ocone Formula

Why is the Clark-Ocone formula: $F = E[F] + \int_0^T E[D_t F | F_t] dW_t$ important, besides its applications to finance. That is, can you give examples of any important pieces of pure theory where ...
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18 views

Covariance between brownian bridge and its max.

Does anyone know how to compute $\text{Cov}[\max_{s\in [0,1]}B(s), B(t)]$ where $B(t)$ is the standard Brownian bridge on the interval $[0,1]$?
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46 views

i.i.d random variables

Looking at this question I. i. d distributions as best car offers I wonder about the following one: Can we find the distribution function of $X_N$ where, $$ N= \min \{ n \geq 1 \mid X_n > ...
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13 views

stationary probability distribution in markov chains [on hold]

What will be the stationary probability distribution of an absorbing state in a given markov chain?
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18 views

How to Maximize the Probabilty of Doubling Your Money!

This is an interesting questions I have heard mentioned a few times but don't know how to solve. Consider a geometric Brownian motion with some finite time horizon $T$ and a money market account with ...
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15 views

Showing the distribution of a poisson process

A large lump of radioactive material has a long half life. Let $D(t)$ be the total number of decays which occur in the radioactive material in the period of $t$ hours starting at noon on a particular ...
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17 views

Stochastic PDE representation

I am trying to find a pde which $u$ satisfies when $u(x) = E^{x}[\cos(X_1)]$ where $dX_t = \sin(nX_t)\,dt + dW_t$ and $X_0 = x$. I have tried using Feynman-Kac but I can't seem to get it into the ...
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30 views

stochastic process with random variable as time, measurable

i have a problem with the following exercise: Let $X$ be an measurable $\mathbb{R}^d$ valued stochastic process on $(\Omega,\mathcal{F},\mathbb{P})$ and $T$ a finite $T : \Omega\to \big[ 0, +\infty ...
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50 views

Hazard function of $\min(X_1, X_2)$

Suppose I have two random variables, $X_1$ and $X_2$, that are independent (but not identically distributed) and assume both have hazard functions $\lambda_1(s)$ and $\lambda_2(s)$, for $s > 0$. ...
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+100

Poisson process and uniform random variable

Question: A single-pump petrol station is running low on petrol. The total volume of petrol remaining for sale is $100$ litres. Suppose cars arrive to the station according to a Poisson ...
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12 views

Restarting a Markov chain

I'm reading an article and having difficulty understanding some basic stochastic processes (I'm new to the subject so please pardon my wording of the question). Let $S$ be a set of states and $Q$ be a ...
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10 views

restarting a Markov chain

I'm reading an article and having difficulty understanding some basic stochastic processes (I'm new to the subject so please pardon my wording of the question). Let $S$ be a set of states and $Q$ be a ...
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38 views

Ito Integral of a SDE [on hold]

I would like to get help in solving the following Ito stochastic equation: $dY_t=-dW_t \, (Y_t^2+1)$ The process $W_t$ is the standard Brownian motion. Is it possible to get a path solution ...
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1answer
18 views

Branching processes extinction (homework)

This is my stochastic process course homework. I can solve (a)(b), which are easy to prove. But I have no idea about (c). Could you give me some idea?
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60 views

differential equations for continiuos markov processes

I'm trying to find the forward equations for birth-and-death processes with no birth, that is, when all $\lambda$ coefficients are zero. The forward equation for a birth-and-death process has the ...
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22 views

Mean-value like result for stochastic integrals

I'm working through Protter's book on stochastic integration; this is problem 16 from chapter 2. I can't seem to crack it--maybe someone here can give me a hint? Let B be standard Brownian and H be a ...
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1answer
14 views

Computing the PMF for N(t) in a renewal process

I'm in a stochastic processes course, and we just started on renewal theory. Unfortunately, we skipped the section on queuing theory, and nearly example in my textbook for renewal processes uses ...
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25 views

Computing joint probability [closed]

Let $X,Y\sim \text{Exp}(1)$ (exponential random variables with parameter $1$). Then prove that $$Pr(X> z_1, \frac{Y}{X} > z_2) = \dfrac{e^{-z_1 (1+z_2)}}{1+z_2}, \forall z_1,z_2>0$$
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+100

Are these statements of my professor about periodicity of harmonic processes in time series analysis correct?

Assume $X_t$ is a harmonic stochastic process, i.e., $$X_t = \sum_{j=-k}^k A_j \exp(i \lambda_j t)$$ where the frequencies $\lambda_j$ are given and $A_j$ are uncorrelated random variables with zero ...
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Is the sum of two compensated poisson processes always a martingale?

Let $M^{1}_t=N^{1}_t-t\lambda^{1}$ and $M^{2}_t=N^{2}_t-t\lambda^{2}$ be two compensated poisson processes, where $\lambda^{1}$ and $\lambda^{2}$ are the constant intensities of $N^{1}_t$ and ...
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21 views

Application of Ito's Lemma to integral expression

I have a problem applying Ito's lemma. I know that if: $dX_t= \mu_t \, dt + \sigma_t \, dB_t$ then for $f(t,x)$: $df(t,X_t) =\left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial ...
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24 views

Probabilistic model of parallel web servers

Note: The following probabilistic model of parallel web servers is abstracted from an engineering project. I am not good at probability theory and I am seeking some evaluations and suggestions. ...
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6 views

Counting processes question

Let's say that arrivals at a counter come at times of a Poisson process with rate $\lambda$. A ball that arrives to an unlocked counter is registered and then locks the counter for an amount of time ...
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Ito's lemma on $a=\int_0^t x(q) \mathrm{d}B(q)$,where $B=$brownian motion process. [closed]

Can someone help me apply Ito's lemma on $a=\int_0^t x(q) \mathrm{d}B(q)$, where $B=$brownian motion process. I did this so far: $$\mathrm{d} a=\frac{\partial }{\partial B} \bigg[ \int_0^t x(q) ...
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Waiting time probability question

I want to solve the following problem: A dentist works 4 hours a day. Patients arrive on the average of 1 per 20 minutes and one patient spends on average 15 minutes with the dentist. Both time ...
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29 views

Poisson process and moment generating function

If we have a Poisson process $ \lbrace N(t), t > 0 \rbrace $ with rate $\lambda > 0 $ and if we have a random variable $S$ having a uniform distribution on the interval $[0,2]$. I was wondering ...
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35 views

Construct SDE with two uncorrelated Brownian motions

Using $Y(t) = wX_1(t) + \sqrt{1-w^2}X_2(t)$ as a model to construct a process where X1 and X2 are brownian motions with drifts and brownian increments $dX_1(t)= \mu_1dt + \sigma_1dW_1(t)$ ...
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16 views

Covariance function meaning

I have this sentence in a report but I don't quiet know what it means. I am familier with covariance and covariance matrices but not with covariance functions. $f(t)$ is a continuous-time ...
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9 views

Differentiability of random processes.

I know the appropriate criterions for mean-square differentiability of random processes. These criterions are connected with covariance function of a process. Are there any criterions for ...
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Does this Stochastic Differential Equation have a name?

I came across this SDE and since I am not an expert I am wondering if this SDE is known to have an closed form solution for first passage times. The SDE is $$dY_t=(a+be^{ct}) \, dt+\sigma \, dB_t$$ ...
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19 views

Branching processes, extinction probability

Why do we assume that a branching process starts with one ancestor. What happens to extinction probability if we have more than one ancestor in generation Y(0)?
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Ornstein-Uhlenbeck process written explicitly

I need to show that the Ornstein-Uhlenbeck process, $$ dX_t = -\theta X_tdt + dB(t) $$ Where $X_0=0$, $B(t)$ is Brownian motion and $\theta>0$ can be written explicitly as: $$ X_t=B(t) - \theta ...
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First property of discrete time homogenous markov chain

I'm trying to understand the properties of a DTHMC. I am having trouble understanding with the first one. My textbook says - "$X_t$ takes values in $X$ for all $t$ (i.e. $X_t$ is a random variable ...
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16 views

Showing a process is a martingale from Ito's lemma

Suppose that we have the process $t-W^2(t)$ where $W(t)$ is a Brownian motion with filtration $\mathcal{F}_t$. It is easy to show that this is a martingale by computing ...
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Solving the SDE $dX_t=bdt+cX_t dW_t$

I want to solve the SDE $dX_t=bdt+cX_t dW_t$, $X_0=0$ for $b,c\in\mathbb R$. I start by rewriting this as $$dX_t=(\mu_1+\mu_2 X_t )dt+(\sigma_1+\sigma_2 X_t )dW_t$$ where $\mu_1=b, \mu_2=0, ...
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+50

Is there a standard proof for $\mathbb P(S^X_n\text{ hits }A\text{ before }B) >\mathbb P(S^Y_n\text{ hits }A\text{ before }B)$?

Let $X_i$ and $Y_i$ be two continuous random variables on $\mathbb{R}$ having distribution functions $F$ and $G$, respectively satisfying $G(y)>F(y)$ for all $y$. Let futhermore $S^X_n=\sum_{i=1}^n ...
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Poisson process: Has my book used a necessary condition, when it should have used a sufficient condition?

My book says that if we know that if we are viewing a poisson process with length $t$, and know that $n$ events happened in that interval, than the time that any of those events happened is uniformly ...
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Variance of exit time for simple symmetric random walk

For a simple symmetric random walk starting at 0 (that is, a Markov chain on the integers starting at 0 with equal probabilities of going to the left and right at each step), I want to compute the ...
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1answer
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M/M/1 queue with probability of new client leaving

I'm looking at a M/M/1 queue system and trying to show that $\{M_t\}_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda_i = ...
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arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
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Showing the square of a Markov process is or isn't Markov

Hi I am trying to show that if $X_n$ is a markov process, whether or not $X_n^2$ is a markov process. $X_n$ is a markov process if $P\{X_k = a_k|X_{k-1} = a_{k-1}, X_{k-2} = a_{k-2}, ..., X_k = a_1 ...
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26 views

How are the waiting times distributed, poisson process.

I am wondering how the waiting times are distributed for the poisson process, conditioned on a number of events by time t. Look at this theorem: Here, the S's are the sum of the waiting time to ...
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37 views

Solution to a stochastic differential equation

I could really do with some help on this question, have no idea where to start. Any advice would be much appreciated, thank u in advance. I am given $$\begin{align}dx(t)&=(1+x(t))dt + x(t) ...
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48 views

Find the distribution of the maximum of a Wiener Process with negative drift

So.. what I have now is Let $M=max\{W_t; 0\leq t <\infty\}$ since $W_0=0$, $M\geq 0$ with probability 1. So, $P(M>x)=P(T_x<\infty)$ where $T_x$ is the stopping time, so we now use the ...
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42 views

Markovian birth-death process [closed]

A linear Markovian death process, initialized at five members, experiences an average daily death rate $u=0.1$. Determine the probability of having fewer than three members in the population after a ...